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為 3d 無人機添加角度

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為 3d 無人機添加角度

如何將歐拉角和 XYZ 角添加到程式碼中，如圖所示？在此輸入影像描述

\documentclass[tikz]{standalone}
%\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{3d, arrows.meta, calc, quotes}
\begin{document}
\begin{tikzpicture}[
  x = ( 20:-2cm), y = (-30: 2cm), z = ( 80: 1.6cm),
  > = Triangle]
\draw coordinate (O) (0,0,-.05) ellipse [radius=.2];

\foreach \xy/\yx in {x/y, y/x} {
  \fill[white] (xyz cs: \xy = -1, \yx = -.05) -- (xyz cs: \xy =  1, \yx=-.05)
            -- (xyz cs: \xy =  1, \yx =  .05) -- (xyz cs: \xy = -1, \yx= .05) -- cycle;
  \foreach \lmr in {-.05, 0, .05}
    \draw[gray] (xyz cs: \xy=-1, \yx = \lmr) -- (xyz cs: \xy=1, \yx=\lmr);
  \draw[fill=white, radius=.15] (xyz cs: \xy = -1) ellipse[]
                                (xyz cs: \xy =  1) ellipse[];
}
\draw[fill=white] (O) ellipse [radius=.2];

\foreach \xyz/\Label/\Ang in {% \Ang needs to start with a ,
  x/left/ {, longitude = 70},
  y/right/{, longitude =-20},
  z/above/{, longitude = 70, latitude=70}%
} {
  \draw[->, densely dashed, red] (0, 0, 0) -- ++(xyz cs: \xyz= 1.5)
    coordinate (ax-\xyz) node[\Label] {$\xyz$};
  \draw[blue, ->]
     (0, 0, 0) -- (xyz spherical cs: radius = 1.5,/tikz/style/.expanded=\Ang)
      coordinate (ax-\xyz') node[\Label] {$\xyz$};
}

\draw[canvas is xy plane at z=0]
  (0, 1) arc[start angle=90, delta angle=20, radius=1]
  node[midway, below right] {$\varphi$};
\end{tikzpicture}
\end{document}

答案1

我建議使用該tikz-3dplot包。在包內，已經有一個使用歐拉角定義的座標變換。您可以找到類似的問題並提供一些有用的答案這裡。基本上，您可以定義一個主座標系，您可以在其中設定兩個角度的方向（例如\tdplotsetmaincoords{50}{115}）。然後，可以透過命令旋轉該座標系\tdplotsetrotatedcoords{<angle1>}{<angle2>}{<angle3>}。在該套件中，此指令的實作是使用序列 zyz 中的歐拉角執行座標系的旋轉。

要獲得另一個序列，可以重新定義轉換命令，如已討論的這和這個答案到前面提到的問題。這樣，您最終可以添加所有可能的序列並在它們之間進行切換。在下面的程式碼中，我使用了 zyx 序列（稱為 Tait-Bryan 或 Cardan 角落（請參閱歐拉角的維基百科))，因為它最接近給定的圖片，也是最常見的 Tait-Byran 角，也是 zxz 序列作為最常見的經典歐拉角。

然後，您可以在主座標系或旋轉座標系中進行繪製。這樣，就可以輕鬆添加任何 3D 繪圖，例如無人機以及用於標記角度的弧線。軟體tikz-3dplot包包含一個命令，但我發現使用您已經使用過的\tdplotdrawarc[coordinates_frame,->,colo]{origin}{radius}{start angle}{end angle}{node info}{label}選項在旋轉座標系的平面上手動繪製圓弧更直觀。canvas is xy plane at z=0

\documentclass[tikz]{standalone}
%\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows.meta, quotes}
%\usetikzlibrary{3d, arrows.meta, calc, quotes}

\usepackage{tikz-3dplot}

% Define new arc command
% Syntax: [draw options] (center) (initial angle:final angle:radius)
% https://tex.stackexchange.com/a/66220
\def\centerarc[#1](#2)(#3:#4:#5){%
    \draw [#1] ($(#2)+({#5*cos(#3)}, {#5*sin(#3)})$) arc (#3:#4:#5)%
}
 
% Redefine the rotation sequence for the tikz3d-plot to Euler-Angles:
% z-x-z with alpha-beta-gamma (psi-theta-phi)
% https://tex.stackexchange.com/q/118069/98906
\newcommand{\tdseteulerzxz}{%
    \renewcommand{\tdplotcalctransformrotmain}{%
        %
        % Determine the sin and cos of the specified angle in degrees
        % \tdplotsinandcos{sin}{cos}{theta}
        % - #1: Returns sin(#3)
        % - #2: Returns cos(#3)
        % - #3: User-specified angle theta
        \tdplotsinandcos{\sinalpha}{\cosalpha}{\tdplotalpha} 
        \tdplotsinandcos{\sinbeta}{\cosbeta}{\tdplotbeta}
        \tdplotsinandcos{\singamma}{\cosgamma}{\tdplotgamma}
        %
        % Define trigonometric abbreviations
        %
        \tdplotmult{\sasb}{\sinalpha}{\sinbeta}
        \tdplotmult{\sacb}{\sinalpha}{\cosbeta}
        \tdplotmult{\sacbsg}{\sacb}{\singamma}
        \tdplotmult{\sacbcg}{\sacb}{\cosgamma}
        \tdplotmult{\sasg}{\sinalpha}{\singamma}
        \tdplotmult{\sacg}{\sinalpha}{\cosgamma}
        %
        \tdplotmult{\sbsg}{\sinbeta}{\singamma}
        \tdplotmult{\sbcg}{\sinbeta}{\cosgamma}
        %
        \tdplotmult{\casb}{\cosalpha}{\sinbeta}
        \tdplotmult{\cacb}{\cosalpha}{\cosbeta}
        \tdplotmult{\cacbsg}{\cacb}{\singamma}
        \tdplotmult{\cacbcg}{\cacb}{\cosgamma}
        \tdplotmult{\casg}{\cosalpha}{\singamma}
        \tdplotmult{\cacg}{\cosalpha}{\cosgamma}
        %
        % Define the entries for the rotation matrix from the B-System to the I-System
        % This is A_IB = (A_BI)^T
        %
        \pgfmathsetmacro{\raaeul}{+\cacg - \sacbsg}
        \pgfmathsetmacro{\rabeul}{-\casg - \sacbcg}
        \pgfmathsetmacro{\raceul}{+\sasb}
        %
        \pgfmathsetmacro{\rbaeul}{+\sacg + \cacbsg}
        \pgfmathsetmacro{\rbbeul}{-\sasg + \cacbcg}
        \pgfmathsetmacro{\rbceul}{-\casb}
        %
        \pgfmathsetmacro{\rcaeul}{+\sbsg}
        \pgfmathsetmacro{\rcbeul}{+\sbcg}
        \pgfmathsetmacro{\rcceul}{+\cosbeta}
        %
    }
}
 
% Redefine the rotation sequence for the tikz3d-plot to Cardan-Angles:
% z-y-x with alpha-beta-gamma
% https://tex.stackexchange.com/q/118069/98906
\newcommand{\tdsetcardanzyx}{%
    \renewcommand{\tdplotcalctransformrotmain}{%
        %
        % Determine the sin and cos of the specified angle in degrees
        % \tdplotsinandcos{sin}{cos}{theta}
        % - #1: Returns sin(#3)
        % - #2: Returns cos(#3)
        % - #3: User-specified angle theta
        \tdplotsinandcos{\sinalpha}{\cosalpha}{\tdplotalpha} 
        \tdplotsinandcos{\sinbeta}{\cosbeta}{\tdplotbeta}
        \tdplotsinandcos{\singamma}{\cosgamma}{\tdplotgamma}
        %
        % Define trigonometric abbreviations
        %
        \tdplotmult{\sasb}{\sinalpha}{\sinbeta}
        \tdplotmult{\sasbsg}{\sasb}{\singamma}
        \tdplotmult{\sasbcg}{\sasb}{\cosgamma}
        %
        \tdplotmult{\sacb}{\sinalpha}{\cosbeta}
        \tdplotmult{\sasg}{\sinalpha}{\singamma}
        \tdplotmult{\sacg}{\sinalpha}{\cosgamma}
        %
        \tdplotmult{\casb}{\cosalpha}{\sinbeta}
        \tdplotmult{\casbsg}{\casb}{\singamma}
        \tdplotmult{\casbcg}{\casb}{\cosgamma}
        %
        \tdplotmult{\cacb}{\cosalpha}{\cosbeta}
        \tdplotmult{\casg}{\cosalpha}{\singamma}
        \tdplotmult{\cacg}{\cosalpha}{\cosgamma}
        \tdplotmult{\cbsg}{\cosbeta}{\singamma}
        \tdplotmult{\cbcg}{\cosbeta}{\cosgamma}
        %
        % Define the entries for the rotation matrix from the B-System to the I-System
        % This is A_IB = (A_BI)^T
        %
        \pgfmathsetmacro{\raaeul}{+\cacb}
        \pgfmathsetmacro{\rabeul}{+\casbsg - \sacg}
        \pgfmathsetmacro{\raceul}{+\sasg+\casbcg}
        %
        \pgfmathsetmacro{\rbaeul}{+\sacb}
        \pgfmathsetmacro{\rbbeul}{+\cacg + \sasbsg}
        \pgfmathsetmacro{\rbceul}{\sasbcg-\casg}
        %
        \pgfmathsetmacro{\rcaeul}{-\sinbeta}
        \pgfmathsetmacro{\rcbeul}{+\cbsg}
        \pgfmathsetmacro{\rcceul}{+\cbcg}
        %
    }
}
 
% Plot display orientation
\tdplotsetmaincoords{50}{115}

\begin{document}


% z-y-x Tait-Bryan angles
\begin{tikzpicture}[scale=2,tdplot_main_coords,> = Triangle]
    % change ratation to z-y-x Tait-Bryan / Cardan angles
    \tdsetcardanzyx
    
    \pgfmathsetmacro{\angpsi}{-20}  % or alpha
    \pgfmathsetmacro{\angtheta}{-10}    % or beta
    \pgfmathsetmacro{\angphi}{-5}   % or gamma

    \pgfmathsetmacro{\r}{2} % unit length radius      
    
    \coordinate (O) at (0,0,0);     
    
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{\angphi}    
    \begin{scope}[tdplot_rotated_coords,scale=1.2]
        \draw coordinate (O) (0,0,-.05) ellipse [radius=.2];
        \foreach \xy/\yx in {x/y, y/x} {
          \fill[white] (xyz cs: \xy = -1, \yx = -.05) -- (xyz cs: \xy =  1, \yx=-.05)
                    -- (xyz cs: \xy =  1, \yx =  .05) -- (xyz cs: \xy = -1, \yx= .05) -- cycle;
          \foreach \lmr in {-.05, 0, .05}
            \draw[gray] (xyz cs: \xy=-1, \yx = \lmr) -- (xyz cs: \xy=1, \yx=\lmr);
          \draw[fill=white, radius=.15] (xyz cs: \xy = -1) ellipse[]
                                        (xyz cs: \xy =  1) ellipse[];
        }
        \draw[fill=white] (0,0,0) ellipse [radius=.2];
        
    \end{scope}     
    
    
    % original coordinate system
    \foreach \xyz/\Label in {x/left,y/right,z/above}
        \draw[->,blue] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz$};
    
    % angle of first rotation around z-axis
    \draw[canvas is xy plane at z=0,->] (1.5,0) arc (0:\angpsi:1.5);
    \node at ({1.7*cos(\angpsi/2)},{1.7*sin(\angpsi/2)},0){$\psi$};
    \draw[canvas is xy plane at z=0,->] (0,1.5) arc (90:90+\angpsi:1.5);
    \node at ({-1.7*sin(\angpsi/2)},{1.7*cos(\angpsi/2)},0){$\psi$};    
    \tdplotsetrotatedcoords{\angpsi}{0}{0}
    
    % new coordinate system
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {x/left,y/right}
            \draw[->,green] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz'$};
        
        % angle of second rotation around new y-axis
        \draw[canvas is xz plane at y=0,->] (1.5,0) arc (0:-\angtheta:1.5);
        \node at ({1.7*cos(\angtheta/2)},0,{-1.7*sin(\angtheta/2)}){$\theta$};
        \draw[canvas is xz plane at y=0,->] (0,1.5) arc (90:90-\angtheta:1.5);
        \node at ({1.7*sin(\angtheta/2)},0,{1.7*cos(\angtheta/2)}){$\theta$};
    \end{scope} 
    
        
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{0}  
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {z/above}
            \draw[->,magenta] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz''$};
        
        % angle of third rotation around new x-axis
        \draw[canvas is yz plane at x=0,->] (1.5,0) arc (0:\angphi:1.5);
        \node at (0,{1.7*cos(\angphi/2)},{1.7*sin(\angphi/2)}){$\varphi$};
        \draw[canvas is yz plane at x=0,->] (0,1.5) arc (90:90+\angphi:1.5);
        \node at (0,{-1.7*sin(\angphi/2)},{1.7*cos(\angphi/2)}){$\varphi$};
    \end{scope}     
    
    % rotated coordinate system
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{\angphi}    
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {x/left,y/right,z/above}
            \draw[->,densely dashed,red] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\tilde\xyz$};
    \end{scope}
\end{tikzpicture}

% z-x-z Euler angles
\begin{tikzpicture}[scale=2,tdplot_main_coords,> = Triangle]
    % change ratation to z-y-x Tait-Bryan / Cardan angles
    \tdseteulerzxz
    
    \pgfmathsetmacro{\angpsi}{-10}  % or alpha
    \pgfmathsetmacro{\angtheta}{-20}    % or beta
    \pgfmathsetmacro{\angphi}{-10}  % or gamma

    \pgfmathsetmacro{\r}{2} % unit length radius      
    
    \coordinate (O) at (0,0,0);     
    
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{\angphi}    
    \begin{scope}[tdplot_rotated_coords,scale=1.2]
        \draw coordinate (O) (0,0,-.05) ellipse [radius=.2];
        \foreach \xy/\yx in {x/y, y/x} {
          \fill[white] (xyz cs: \xy = -1, \yx = -.05) -- (xyz cs: \xy =  1, \yx=-.05)
                    -- (xyz cs: \xy =  1, \yx =  .05) -- (xyz cs: \xy = -1, \yx= .05) -- cycle;
          \foreach \lmr in {-.05, 0, .05}
            \draw[gray] (xyz cs: \xy=-1, \yx = \lmr) -- (xyz cs: \xy=1, \yx=\lmr);
          \draw[fill=white, radius=.15] (xyz cs: \xy = -1) ellipse[]
                                        (xyz cs: \xy =  1) ellipse[];
        }
        \draw[fill=white] (0,0,0) ellipse [radius=.2];
        
    \end{scope}     
    
    
    % original coordinate system
    \foreach \xyz/\Label in {x/left,y/right,z/above}
        \draw[->,blue] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz$};
    
    % angle of first rotation around z-axis
    \draw[canvas is xy plane at z=0,->] (1.5,0) arc (0:\angpsi:1.5);
    \node at ({1.7*cos(\angpsi/2)},{1.7*sin(\angpsi/2)},0){$\psi$};
    \draw[canvas is xy plane at z=0,->] (0,1.5) arc (90:90+\angpsi:1.5);
    \node at ({-1.7*sin(\angpsi/2)},{1.7*cos(\angpsi/2)},0){$\psi$};    
    \tdplotsetrotatedcoords{\angpsi}{0}{0}
    
    % new coordinate system
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {x/left,y/right}
            \draw[->,green] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz'$};
        
        % angle of second rotation around new x-axis
        \draw[canvas is yz plane at x=0,->] (1.5,0) arc (0:\angtheta:1.5);
        \node at (0,{1.7*cos(\angtheta/2)},{1.7*sin(\angtheta/2)}){$\theta$};
        \draw[canvas is yz plane at x=0,->] (0,1.5) arc (90:90+\angtheta:1.5);
        \node at (0,{-1.7*sin(\angtheta/2)},{1.7*cos(\angtheta/2)}){$\theta$};
    \end{scope} 
    
        
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{0}  
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {y/right}
            \draw[->,magenta] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\xyz''$};
        
        % angle of third rotation around new z-axis
        \draw[canvas is xy plane at z=0,->] (1.5,0) arc (0:\angphi:1.5);
        \node at ({1.7*cos(\angphi/2)},{1.7*sin(\angphi/2)},0){$\varphi$};
        \draw[canvas is xy plane at z=0,->] (0,1.5) arc (90:90+\angphi:1.5);
        \node at ({-1.7*sin(\angphi/2)},{1.7*cos(\angphi/2)},0){$\varphi$};
    \end{scope}     
    
    % rotated coordinate system
    \tdplotsetrotatedcoords{\angpsi}{\angtheta}{\angphi}    
    \begin{scope}[tdplot_rotated_coords]
        \foreach \xyz/\Label in {x/left,y/right,z/above}
            \draw[->,densely dashed,red] (O) -- (xyz cs: \xyz=\r) node[\Label]{$\tilde\xyz$};
    \end{scope}
\end{tikzpicture}


\end{document}

結果：

泰特·布萊恩·齊克斯 Tait-Bryan_zyx 歐拉zxz 歐拉_zxz

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